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Mirrors > Home > MPE Home > Th. List > oduleval | Structured version Visualization version GIF version |
Description: Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | ⊢ 𝐷 = (ODual‘𝑂) |
oduval.l | ⊢ ≤ = (le‘𝑂) |
Ref | Expression |
---|---|
oduleval | ⊢ ◡ ≤ = (le‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6909 | . . . . 5 ⊢ (le‘𝑂) ∈ V | |
2 | 1 | cnvex 7933 | . . . 4 ⊢ ◡(le‘𝑂) ∈ V |
3 | pleid 17351 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
4 | 3 | setsid 17180 | . . . 4 ⊢ ((𝑂 ∈ V ∧ ◡(le‘𝑂) ∈ V) → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
5 | 2, 4 | mpan2 689 | . . 3 ⊢ (𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
6 | 3 | str0 17161 | . . . 4 ⊢ ∅ = (le‘∅) |
7 | fvprc 6888 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (le‘𝑂) = ∅) | |
8 | 7 | cnveqd 5878 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ◡∅) |
9 | cnv0 6147 | . . . . 5 ⊢ ◡∅ = ∅ | |
10 | 8, 9 | eqtrdi 2781 | . . . 4 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ∅) |
11 | reldmsets 17137 | . . . . . 6 ⊢ Rel dom sSet | |
12 | 11 | ovprc1 7458 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
13 | 12 | fveq2d 6900 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) = (le‘∅)) |
14 | 6, 10, 13 | 3eqtr4a 2791 | . . 3 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
15 | 5, 14 | pm2.61i 182 | . 2 ⊢ ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
16 | oduval.l | . . 3 ⊢ ≤ = (le‘𝑂) | |
17 | 16 | cnveqi 5877 | . 2 ⊢ ◡ ≤ = ◡(le‘𝑂) |
18 | oduval.d | . . . 4 ⊢ 𝐷 = (ODual‘𝑂) | |
19 | eqid 2725 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
20 | 18, 19 | oduval 18283 | . . 3 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
21 | 20 | fveq2i 6899 | . 2 ⊢ (le‘𝐷) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
22 | 15, 17, 21 | 3eqtr4i 2763 | 1 ⊢ ◡ ≤ = (le‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 〈cop 4636 ◡ccnv 5677 ‘cfv 6549 (class class class)co 7419 sSet csts 17135 ndxcnx 17165 lecple 17243 ODualcodu 18281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-dec 12711 df-sets 17136 df-slot 17154 df-ndx 17166 df-ple 17256 df-odu 18282 |
This theorem is referenced by: oduleg 18285 odupos 18323 oduposb 18324 odulub 18402 oduglb 18404 posglbdg 18410 oduprs 32780 odutos 32784 mgccnv 32815 ordtcnvNEW 33649 ordtrest2NEW 33652 glbprlem 48167 |
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